Aeronautical Engineering is a sub-branch of
Aerospace Engineering. Aerospace Engineering is the science and
engineering of flying vehicles operating within and outside of the
earth’s atmosphere while Aeronautical engineering is the science and
engineering of flying vehicles operating within the Earth’s atmosphere
only like aircrafts.

An Aeronautical Engineering degree would be
useful if you wish to have a career only related to aircrafts, helicopters
and related fields. An Aerospace engineer besides working on aircrafts,
helicopters can also work on rockets, missiles, space station,
space-shuttle, lunar or interplanetary probes and more. I would suggest
pursuing a career in Aerospace engineering as it gives far greater
opportunities than an Aeronautical engineering career.